Multiplication between Hardy spaces and their dual spaces

Abstract

For any p∈(0,1) and α=1/p−1, let Hp(Rn) and Cα(Rn) be the Hardy and the Campanato spaces on the n-dimensional Euclidean space Rn, respectively. In this article, the authors find suitable Musielak–Orlicz functions Φp, defined by setting, for any x∈Rn and t∈[0,∞),Φp(x,t):={t1+[t(1+|x|)n]1−pwhenn(1/p−1)∉N,t1+[t(1+|x|)n]1−p[log⁡(e+|x|)]pwhenn(1/p−1)∈N, and then establish a bilinear decomposition theorem for multiplications of functions in Hp(Rn) and its dual space Cα(Rn). To be precise, for any f∈Hp(Rn) and g∈Cα(Rn), the authors prove that the product of f and g, viewed as a distribution, can be decomposed into S(f,g)+T(f,g), where S is a bilinear operator bounded from Hp(Rn)×Cα(Rn) to L1(Rn) and T a bilinear operator bounded from Hp(Rn)×Cα(Rn) to the Musielak–Orlicz Hardy space HΦp(Rn) associated with the above Musielak–Orlicz function Φp. Such a bilinear decomposition is sharp when nα∉N, in the sense that any other vector space Y⊂HΦp(Rn) adapted to the above bilinear decomposition should satisfy (L1(Rn)+Y)⁎=(L1(Rn)+HΦp(Rn))⁎. To obtain the sharpness, the authors establish a characterization of the class of pointwise multipliers of Cα(Rn) by means of the dual space of HΦp(Rn), which is of independent interest. As an application, an estimate of the div-curl product involving the space HΦp(Rn) is discussed. This article naturally extends the known sharp bilinear decomposition of H1(Rn)×BMO(Rn).